Distance Functions, Critical Points, and the Topology of Random v{C}ech Complexes

For a finite set of points $P$ in $R^d$, the function $d_P: R^d \to R^+$ measures Euclidean distance to the set $P$. We study the number of critical points of $d_P$ when $P$ is a Poisson process. In particular, we study the limit behavior of $N_k$ - the number of critical points of $d_P$ with Morse index $k$ - as the density of points grows. We present explicit computations for the normalized, limiting, expectations and variances of the $N_k$, as well as distributional limit theorems. We link these results to recent results in which the Betti numbers of the random \v{C}ech complex based on $P$ were studied.